The integral monodromy of hyperelliptic and trielliptic curves

Achter, Jeffrey D.; Pries, Rachel

doi:10.1007/s00208-006-0072-0

Cited by 31 publications

(71 citation statements)

References 23 publications

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“…So we must rule out the case where K 2 is isomorphic to S and is diagonally embedded in Ω S. We will do this by exhibiting a nontrivial element of K 2 with trivial projection to one factor. By Lemma 5, β 3 1 acts trivially on V . On the other hand, β 3 1 is G-conjugate to β 3 b−1 , whose image in PSp(V ) is nontrivial, by the same lemma.…”

Section: Lemmamentioning

confidence: 94%

Monodromy groups of Hurwitz-type problems

Allcock

Hall

2010

Advances in Mathematics

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We solve the Hurwitz monodromy problem for degree 4 covers. That is, the Hurwitz space H 4,g of all simply branched covers of P 1 of degree 4 and genus g is an unramified cover of the space P 2g+6 of (2g + 6)-tuples of distinct points in P 1 . We determine the monodromy of π 1 (P 2g+6 ) on the points of the fiber. This turns out to be the same problem as the action of π 1 (P 2g+6 ) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, 3 N , in place of Z/2. This in turn allows us to answer a question of Ellenberg concerning families of Galois covers of P 1 with deck group (Z/N ) 2 :S 3 .A ramified cover C of P 1 of degree d is said to have simple branching if the fiber over every branch point has d − 1 distinct points. Another way to say this is that for each branch point p, the permutation of the sheets of the cover induced by a small loop around p is a transposition, i.e., a permutation of cycle-shape 2 1 1 d−2 . An Euler characteristic argument (or the Hurwitz formula)

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Section: Lemmamentioning

confidence: 94%

Monodromy groups of Hurwitz-type problems

Allcock

Hall

2010

Advances in Mathematics

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“…(This has been proved independently by Yu (unpublished; see [5,Ex. 2.4]); by the first author and Pries [3,Theorem 3.4]; and by Hall [9,Theorem 4.1]. )…”

Section: Remark 24mentioning

confidence: 99%

On the probability of having rational $$\ell$$ -isogenies

Achter

Sadornil

2008

Arch. Math.

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“…Since Φ ℓ is conjecturally semisimple and our methods only handle semisimple representations, we denote, for all ℓ, the semisimplification of Φ ℓ by Φ ss ℓ . We say that {Φ ss ℓ } ℓ is the semisimplification of the system (1). Let Γ ℓ and G ℓ be respectively the monodromy group (Galois image) and the algebraic monodromy group of Φ ss ℓ .…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.2. Let {Φ ℓ } ℓ be the system (1) and G ℓ the connected algebraic monodromy group of Φ ss ℓ for all ℓ. Suppose Hypothesis A is satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…For the A 6 case discussed above, Corollary 1.5 implies (by studying the Gal Q action on the Dynkin diagram of G der Q ) either the Chevalley group A 6 (ℓ) occurs for ℓ ≫ 0 or there is a quadratic extension F of Q such that for ℓ ≫ 0, the Chevalley group A 6 (ℓ) occurs for ℓ that splits completely and the Steinberg group 2 A 6 (ℓ 2 ) occurs for ℓ that is inert. Such a congruence is useful to the inverse Galois problem and appears, for example, in the computation of the geometric Z/ℓZ-monodromy of the moduli space of trielliptic curves [1,Theorem 3.8].…”

Section: Introductionmentioning

confidence: 99%

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On the rationality of certain type A Galois representations

Hui

2018

Trans. Amer. Math. Soc.

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Abstract. Let X be a complete smooth variety defined over a number field K and i an integer. The absolute Galois group Gal K of K acts on the ithétale cohomology group H í et (XK, Q ℓ ) for all primes ℓ, producing a system of ℓ-adic representations {Φ ℓ } ℓ . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ admits a common reductive Q-form for all ℓ if X is projective. Denote by Γ ℓ and G ℓ respectively the monodromy group and the algebraic monodromy group of Φ ss ℓ , the semisimplification of Φ ℓ . Assuming that G ℓ0 satisfies a group theoretic condition for some prime ℓ 0 (Hypothesis A), we construct a connected quasi-split Q-reductive group G Q which is a common Q-form of G• ℓ for all sufficiently large ℓ. Let G sc Q be the universal cover of the derived group of G Q . As an application, we prove that the monodromy group Γ ℓ is big in the sense that Γ sc ℓ ∼ = G sc Q (Z ℓ ) for all sufficiently large ℓ.

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The integral monodromy of hyperelliptic and trielliptic curves

Cited by 31 publications

References 23 publications

Monodromy groups of Hurwitz-type problems

Monodromy groups of Hurwitz-type problems

On the probability of having rational $$\ell$$ -isogenies

On the rationality of certain type A Galois representations

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